Take \(A,B\)  according to lem:unitary-to-difference, with \(A,B\ge 0\) , \(X=A-B\)  and \(|A|+|B|\le \textsf {width}(G)\) . Since \(A,B\ge 0\) , there exist paths \(P_{1},\dots ,P_{|A|+|B|}\)  such that \(A=P_{1}+\dots +P_{|A|}\)  and \(B=P_{|A|+1}+\dots +P_{|A|+|B|}\)  [1]}. Since \(X=A-B\) , we can write \(X=P_{1}+\dots +P_{|A|}-P_{|A|+1}-\dots -P_{|A|+|B|}\) . Finally, recall that \(|A|+|B|\le \textsf {width}(G)\) , concluding
the proof.
